Contact points of convex bodies

Abstract

LetB be a convex body in ℝ n and let ɛ be an ellipsoid of minimal volume containingB. By contact points ofB we mean the points of the intersection between the boundaries ofB and ɛ. By a result of P. Gruber, a generic convex body in ℝ n has (n+3)·n/2 contact points. We prove that for every ɛ>0 and for every convex bodyB ⊂ ℝ n there exists a convex bodyK having $$m \leqslant C(\varepsilon ) \cdot n\log ^3 n$$ contact points whose Banach-Mazur distance toB is less than 1+ɛ. We prove also that for everyt>1 there exists a convex symmetric body Γ ⊂ ℝ n so that every convex bodyD ⊂ ℝ n whose Banach-Mazur distance to Γ is less thant has at least (1+c 0/t 2)·n contact points for some absolute constantc 0. We apply these results to obtain new factorizations of Dvoretzky-Rogers type and to estimate the size of almost orthogonal submatrices of an orthogonal matrix.